Torsional wave, fourier type, mechanical amplitude transformer



24, 1965 E. EISNER 3,252,335

TORSI ONAL WAVE, FOURIER TYPE, MECHANICAL AMPLITUDE TRANSFORMER med Oct. 3. 1963 :5 Sheets-Sheet 2 Q, R 4 g 22 3 9 l0 z m E (r 5 2 Q) 2 I I x l l l l 1 2 5 IO 20 50 I00 DIAMETER RAT/O, R

E. EISNER 3,252,335

FOURIER TYPE, MECHANICAL AMPLITUDE TRANSFORMER May 24, 1966 TORSIONAL WAVE,

Z5 Sheets-Sheet 5 Filed Oct.

DIAMETER RAT/O I l m \0 c po/ssolvs PAT/O 0.3

l 1 IO I02 FL EXURA L COMPL #1 N05 United States This invention relates to elongated tapered mechanical vibratory members for increasing the magnitude of repetitive physical displacements. More particularly, it relates to members designed for use with torsional vibratory energy.

This application is a continuation-in-part of applicants earlier copending application entitled, Mechanical Amplitude Transformer, Serial No. 267,718, filed March 25, 1963, now United States Patent No. 3,175,406, and assigned to applicants present assignee. The. said earlier application discloses elongated amplitude transformers tapered in accordance with truncated Fourier series of the third and fourth orders to effect magnification of linear longitudinally vibratory motion. The present application extends the theory and practice of the said earlier application to Fourier tapered mechanical amplitude transformers for effecting the magnification of torsional vibratory motion. While the physical dimensions and longitudinal contours of devices of the present invention are distinctly different from those of devices adapted to use longitudinally vibratory energy of like frequency and magnification, the mathematical analysis and underlying theoretical aspects have a common basis as will become apparent hereinunder. Accordingly, applicants abovementioned earlier application is incorporated by reference and made an integral portion of the present application insofar as it is pertinent.

It will be demonstrated hereinbelow that transverse vibrations of the type commonly known as torsional vibrations can be magnified approximately in direct proportion to the 35th power of the ratio of the maximum to the minimum diameters (R of the member rather than being limited, as taught in the prior art for torsional exponentially tapered vibrators to the square (that is, the second power) of this ratio. Furthermore, the appropriate tapered longitudinal contours which afford optimum Fourier mechanical amplitude transformers of torsional vibratory energy from standpoints such as stiffness, and the like, will be presented.

Accordingly, a principal object of the present invention is to extend the capacity and utility of Fourier mechanical amplitude transformers.

A further object is to remove limitations to the stability, ruggedness and case of manufacture of mechanical amplitude transformers for torsional vibratory energy.

Other and further objects. features and advantages of the invention will become apparent from a perusal of the following detailed description of illustrative embodiments of the invention taken in conjunction with the accompanying drawing, wherein:

FIG. 1 illustrates the comparative longitudinal contours for Fourier Torsional and Exponential Torsional types of tapered vibrators both producing an angular ing a linear magnification M of 700;

FIG. 2 illustrates the comparative longitudinal contours for Fourier Torsional, Exponential Torsional, and Fourier Longitudinal tapered vibrators, all three producing a linear magnification M of 700;

FIG. 3 illustrates the comparative angular magnifications versus maximum to minimum diameter ratio, R for Fourier torsional and Exponential torsional types of tapered vibrators;

areas FIG. 4 illustrates the comparative linear magnifications versus diameter ratio, R for Fourier" Torsional, Fourier" Longitudinal and Exponential Torsional or Longitudinal types of tapered vibrators; and

FIG. 5 illustrates the comparative flexural compliances, C versus magnifications for Fourier Torsional (angular magnification), Fourier" Torsional (surface motion magnification) and Fourier Longitudinal types of tapered vibrators.

The equations governing the propagation of longitudinal and torsional waves in thin. tapered rods are of similar form. It is therefore possible by the introduction of appropriate modifications to use substantially the method by which the Fourier type of longitudinally resonant amplitude transformers are designed, as taught in my abovementioned copending application, to design torsionally resonant transformers. These Fourier type torsional transformers are, however, capable of extremely high magnifications which cannot be realized by any practical structure designed in accordance with prior art teachings.

The stark contrast between comparable Fourier type torsional and exponentially tapered torsional transformer contours is graphically illustrated in FIG. 1 of the accompanying drawing in which member 10 is of Fourier type and member 12 is the comparable exponentially tapered type, both being designed for torsional vibrations of like frequency and like angular magnification.

In FIG. 2, the contrast between Fourier torsional, Fourier longitudinal and exponential torsional transformers is illustrated by the comparable contours, 14, 18, and 16, respectively, for like linear or particle velocity magnifications. A magnification of 700 is used for all the transformers in both FIGS. 1 and 2. Incidentally, the crosssectional area of the left end of the exponential torsional transformer 16 is actually less than illustrated since sufficicntly thin lines to illustrate the actual thickness dimension are not practicable for use in the drawing.

The equation governing the (one-dimensional) propagation of torsional waves in thin, straight rods of slowly varying section is readily found by considering the motion of a thin, cross-sectional slice. If 0 is the angular displacement at coordinate x along the rod at time I, and n and p are the shear modulus and the density, respectively, of the (isotropic) material, then l where Dtx) being a chtuz'tctcristic dimension of the cross section and k being nondimensionul functions of crossscctional shape only. K, is sometimes called the torsional rigidity of the section, while K is the second moment of the section about the axis of torsion. K is readily derived by simple integration, while K is given for many shapes in engineering handbooks such as that entitled Formulas for Stress and Strain by R. J. Roark, published by McGraw-Hill Book Company, Inc., New York, 1954.

If the cross section of the rod gets too large, torsional modes not described by Equation 1 can be excited. W. P. Mason discusses this difiiculty for the circular section type of vibrator in his book entitled Physical Acoustics and the Properties of Solids, published by D. Van Nostrand Company, Inc, New York, 1958. The limitation on diametcr" is even more severe for section shapes that depart radically from the circular, such, for example, as narrow rectangles. Also, validity of Equation l requires that the angle of taper of the rod shall not be too great.

If waves of a single frequency, p/21r, where p=21rf, are considered, then so that A, is the wavelength of shear waves of frequency 2/21: in the material. If the shear modulus n is constant, then Also let (Z we d0 25 2 If;

(in It,) l (where in K is the natural logarithm of K sometimes written as log, K This may be reduced to nondimensional (or normalized) form by substituting where I is the resonant length of torsional vibration of the rod to be considered and A is the Wavelength of the torsional waves in an untapered rod of the same crosssectional shape. Then where I! I z U L (7) (X =x/I where I is the longitudinally resonant length, U :u/ 1 where u(x) is the longitudinal displacement at x, A( X) is the cross-sectional area, and Q =21rl where A is the wavelength of extensional waves of the frequency considered in a uniform rod of the material).

The forms of the above Equations 6 and 7 are obviously the same, and it follows that if functions U(Q X), AW X) that together satisfy Equation 7 are found, as taught in my above-mentioned copending application, then functions 0(Q X), K UZ X) of the same forms will sat isfy Equation 6.

It can be shown that, as taught in my copending application, if it is wished to design a longitudinally resonant body having given performance characteristics, it is convenient to find a function U(X) that satisfies the boundary conditions imposed, and then to determine the corresponding shape function, A(X), from Equation 7.

The same approach may clearly be adopted in the torsional case. If due attention is pair to limits, Equation 6 may be inverted to give at 0"(X) 82 0 The torsional problem may differ somewhat from. the longitudinal one in that, while (2 Equation 7, is a constant, SI will be a function of X unless the cross-sectional contour or shape is of constant type (that is, circular, square, or the like) along the length of the body. If the crosssectional contour or shape varied in a predetermined way along the length of the rod, then Q (X) would be known explicitly and in K could be found by integration of Equation 8, given 0(X) and its derivatives, as readily as if SI were constant. The integral would, of course, depend on the arbitrary form of Q The most likely variation of cross-sectional contour or shape would be one in which one dimension were held constant, and all the variation of sectionsize appeared in the other dimension. (One obvious example would be that of rectangular section with one side of the rectangle constant in length.) In that case 9,; would be a function of D(X) and therefore of K (X), so that the integration of Equation 8 would involve appreciably more complexity.

As an arbitrary variation of cross-sectional contour or shape is a rather unlikely requirement to be encountered in practical applications, it will not be exhaustively analyzed at this time. It is felt that the principles of the invention will be more readily perceived if their application is described in connection with rods whose cross-sectional contours or shapes do not vary along their lengths, that is, rods for which Q is constant.

Torsional amplitude transformers For the purposes of the present application, an amplitude transformer" is defined as a mechanically resonant body having two strain-free surfaces on one of which the amplitude of vibration is much bigger than on the other. In my above-mentioned copending application, I have used the semi-inverse approach outlined above to design a class of longitudinally vibrating amplitude transformers, designated as Fourier type, capable of giving large magnification without being unduly slender. It can be seen as demonstrated above that difi'erential equations of the same form govern the longitudinal and torsional designs. Now consider the necessary boundary conditions which, as will presently become apparent, are not identical for the two designs.

The principal difference between the longitudinal and the torsional cases lies in the dependence of the strain (here the shear strain), S, as given by the relation LS]C3D(X)dxlTD(X)0 (X) (9) Where k is another constant dependent on cross-sectional shape only. The engineering definition of shear strain is used, so that k =1 for a rod of circular cross section if D(X) is the radius.

Unlike the strain in the longitudinal case (which is U(X)) this strain depends on D(X). However, since D(X) is finite, the boundary condition S- at a free surface leads to 0'=O, just as it leads to U=0 in the longitudinal case. Using the same arguments as were used in my abovementioned copending application, boundary conditions are imposed on 0(X) of the same form as those used for U(X):

0(0)=0 (10a) 6(1)=M,,00 (10b) o"(1): -M,,n 0 (10f) M, is the magnification required in angular displacement and velocity. Since the governing equation and the boundary conditions for the torsional and longitudinal amplitude transformers are of the same forms, displacement functions, 0(X), U(X) of the same form will solve the problem. Using simpler notation than in my abovementioned copending application,

tional shape only and (I is a chosen constant, (p is also independent of cross-sectional size. It follows, Equation 15, that the maximum particle velocity for given material and maximum strain is similarly independent. Therefore, any

When the values for (X) and the consequent values angular velocity, however large, can be obtained withof (l'tX) and 0"(X) are inserted into Equation 8 the following relation is obtained:

out overstraining by making the cross section small enough. This re-emphasizes the appropriateness of p as For given values of M W and 3 Equation 8 can now be integrated; in fact, the results of the integration in my above-mentioned copending application can be used, since, if

where M, 'y and 11 are the free parameters in the longitudinal case, then K o 'Y'l', 54, X 7:

iU o, 1T, 54, 0) torsional A(M, 7, 0:42, 0) longitudinal (14a) and since K is proportional to D and A 'is proportional to D subject to Conditions 13,

D(U) torsional D(U) longitudinal (14b) Thus, the profiles of the Fourier torsional vibrators may be very simply deduced from those of the Fourier longitudinalv vibrators with the corresponding free parameters. it remains to be seen whether the choice of these parameters made in my above-mentioned copending application is appropriate here.

Optimization It has been shown that in any harmonically resonant body the maximum particle velocity, v is limited by the maximum permissiblestrain. If the maximum strain is S and the speed of sound is c, then (p is a function of geometry and of Poissons ratio of the material. If S and c are suitably defined, (p can often be made independent of Poissons ratio. Furthermore, if

the intuitively obvious definitions of maximum strain and I The maximum particle velocity, v occurs on the surface:

shear waves,

where k, is again a function of cross-sectional shape only.

Therefore, from Equations 9 and 17 It can readily be seen that the ratio of maxima on the right-hand side of Equation 18b is independent of the ab-' solute cross-sectional size, as long as D(X)/D(0) is unchanged. Since k /k and k /k are functions of cross-seca criterion of the suitability of a shape for the production of high amplitude.

In the longitudinal case of my above-mentioned copcnding application the parameters '7 and 11 were varied to provide shapes that were fiexurally stiff and had high values of An examination of the variation of (p and of fiexural stiffness of the torsional designs with w and 3 shows that the samerespective values of the latter parameters provide good compromises. The torsional designs were therefore computed by means of Equation 14b.

Computations In the longitudinal case there was only one displacement, the longitudinal, to consider, and corresponding to' it one magnification, M. In the torsional case, however, there is not only the angular displacement, 0(X), with its magnification, M to consider, but also it is likely to be at least as interesting to consider the surface displacement, u (X), and its magnification, M Mathematically, M, is equivalent to M, and is an input parameter of the problem. On the other hand,

and cannot therefore be found until the profile has been determined.

There are two other quantities that will be tabulated that differ in definition from their longitudinal equivalents, namely (,0 and the impedance.

It can be seen from Equation 18b that (p is a function of cross-sectional shape. However, the quantity (k /k )-(k /l does not differ greatly from 1 for shapes not having sharply re-entrant corners (for which it would be l), and it can only very slightly exceed 1 (1.03 for an equilateral triangle). The choice of crosssectional shape is arbitrary, and so one may tabulate the value of (p for circular cross-section, for'which The impedance, I (X), is defined as the ratio of the couple to the angular velocity at X. It should be noted that this has different dimensions from the impedance in the longitudinal case. It can be shown that where i 2.922 912i 1T 0(0) 0(X) (20b) It follows that a rigid body of polar moment of inertia 1,, may be matched to the vibrator where the body being substituted on the side of increasing X where P (X) is positive and conversely.

From Equation 14!) it can be seen that the torsional vibrator with a given value of M is much less slender than the longitudinal vibrator with M=M,,, with the same length and the same maximum diameter. For instance, if

then the longitudinal vibrators with M=1000 has R =54, TABLE II.PROFILIE$, D sPLACEMENTS AND While the torsional vibrator with M,,=1000 has ze -7.4. 05 335 9 3 it therefore becomes feasible to make and use torsional vibrators with very large values of M Unfortunately, Z DIDO TH/THO BUISUO PT as H increases both the numerator and denominator of the right-hand side of Equation 8, corresponding to Equa- (L LOOOOO 00 0. tion 6a of my above-mentioned copending application, 323? .03301 .0; which are sums and differences of quantities of the order 1 5 1 g g 1,1 of 1, become very small. With M 51000, the normal 0.12052 1.00014 0.00 -0.3321 +01 0.10100 0. 00005 0. 53 -0.50311+01 eight-figure accuracy of the IBM 7090 computer becomes 19429 Q9960. Q34 iradequate, and it becomes necessary to use double pre- 8-2 -8- cision arithmetic. At present this takes the computer 1 5 l g ,1 55 J about ten times as lon a d es sin l recision arithmetic. .3 3 0- 7449 0.54 11452E+0 g S ge 0. 355111 0. 01070 -0.70 0. 200E+01 The convergence of certain iterations 1n the programme Q3885? 0'75865 01651504 become wer th r er M s that the com utation 0.42095 .7 5 3 -3 -107 S 510 e la g O p 0.45333 0.55003 -1.50 0.703104-00 for large values of M,, becomes relatively expensive. (M8572 M2198 AM 0,1671%) For th s reason th 1 r e. value of M that ha here (151810 0-59230 031015-00 b e e 0 e a E St h h 0 f S 2 0.55040 0.57004 -2.37 020110-00 8 1 puted 1S 3 .000, p g i 6 Value 0 n(= 0. 50230 0.55400 -2.55 012113-00 is still quite modest. If specific needs arise for members gglgg; 8-332; 8- 56613-01 having even much larger values of M the computations 2O can, of course, readily be done. lMTheta=m0ml Results 8. 1.00002 1.00 1.00 g. OE+OO .03557 0.0073 0.07 0.07 .73, Table I, hereinbelow, gives the values of the chosen @0731; 0.99262 Q90 0.89 0-154E+01 pflfamelel's, 5 1 4, 'YT, and 0f the Calculated quanfifles, 0.10071 0.00254 0.77 0.70 -0.255E+01 M 5 and the position of the node. The illustrative cases 8-1225; g-ggffig 3-2? 8- 2 :g-gggig} tabulated range from M =5 to M =30,O00 (M =2.7 to 8121041 01037 2 0: 17 0: 10 -gI 170%1 3 1 .25598 0.815 -0. 21 -0. 17 .140 WI -1400). The 5 values in each case lie in the range 029255 068008 A192 0 63 Q315E+01 -0.28 to 0.'35 which is considered a useful range for 032012 056822 4 117134411 0. 30502 0. 43302 -4. 00 -2. 40 0. 510E+00 the purposes described herein. Table II, hereinbelow, (H0226 0.42108 9 36 0 251E 00 gives data for the determination Of the profiles [as 043 33 137412 15 05 00 0,133E-00 D(X)/d(0)], the displacements [as U(X)/U( O)] and g figg 8 3%} 3&3: 5 2; 312%:3 the impedances [as P (X)], as functions of position along 0. 54053 0.20111 -50. 55 -14.7 0. 27513-01 0. 55510 0. 27574 -55. 02 -17. 03 0. 17310-01 the member 2 0.02107 0.20450 -7s. 55 -20.31 0. 10512-01 the positions are all given as distances from the low- 0.55024 0.25700 -s0. 03 -23. 11 0.530%412 0.00401 0.25200 -97.39 -24.00 0.205 -02 amplitude end and expressed in terms of M, where M3138 M5116 doom k 1 2 )vr )i [Mll1ete= 700.00]

0. 1. 00000 1. 00 1. 00 0. In an untapered rod of the same material and cross-sec- 0.03000 0.03330 0.07 0.05 -0.721E+00 tional shape as the amplitude transformer torsional waves 3: $133? 3 3233? 3?: 8: :3: of frequency p/Znwill propogate with wavelength A 0.14075 0.00371 0.50 0.09 -u.3t11E+01 1-1311 0 1 1-3 20 -1.01 0.300. 3 Fourth order FOIU'IC! tors onal amplitude transformers 015083 (M4288 35 4H3 0 ME +01 symbols used m Tables I and II 8.33351 0. 42201 -5. 17 -2. 10 040210-00 020 0. 34437 -14.00 -4.35 017111-00 MTheta=Magnification of angular displacement 8-2822: gggggg :g- 52-3; 8- MSurf=Magnification of surface-particle displacement .41032 8. -10s. 7 0 24.31 gang-g1 .470 .20 -173. 5 -35. 10 .1174 2 Beta4"Free parameter (Equat.1n n) 0.51305 0.113515 -250.47 -47.74 057011-02 LambdaT=Wavelength of torsional waves (Equation 22) 0.55034 0.17304 -352. 30 -01.28 0. 35412-02 samexengm/lambdar 1- 1-110 -10 0 0 .5 .3 .31- Ph1=F1gure of merit for ability to withstand large ampli- 0. 1 141 .12355 ga gs 00.gg 0. snag-g3 03 -1 .12 -02. 0.370-3 mde,(Eq"auns 15 and 0. 73370 0.15017 -700.00 -105.12 -0. Z (Distance from low-amplitude end)/lambdaT r ZNode=Zat displacement node lMThetazsmmml D/Do=Rz1tio of characteristic dimension of cross section at Z to that 5:0 0. 1.00000 1.00 1. 00 -0. TH/TH0=Rat1o of amplitude of angular displacement at 0. 03010 0. 31514 0.110 0. 70 -0.5401 :+00 Zt that at 0. 07237 0. 00730 0.110 0.50 -0.0031i- 00 0.10350 0.33245 0.01 0.54 -0.1441-: 01 SU/SUO=Rat1o of amplitude of surface-particle displace- :1; ment at Z to that at Z v 0' 21711 0' 25707 2 03 4'30 0' 5 2- a. ..2411-01 PT=Reduced impedance at Z (Equation 20b) 0. 25330 0.10105 -03. 00 -15.14 011010-01 Powers of 10 are denoted by E. For example, 1.23E+02 31222:? 8: 1531 32213 8:

and 1.23E-02 mean, respectively, 123. and 0.0123. 0.35100 0.00034 -1530.01 137.30 051111-03 0 ,7. 0 5.1 .1501-03 TABLE I.PARAMETERS AND DERIVED QUANTITIES 8: g; g:

0. 54270 0. 05445 -154 1 1s. 33 -33 04 0. 33410-04 1 M l 1-1311 110113 311311 :30 1- 33:11 7 O 0. 05135 04323 27040. 01 -1304. 20 0. 70111-05 0. 03753 04743 -20233. 53 -1380. 54 0. 3521 1-05 8-88 3- 0. 72372 0.04717 -30000.00 -1414. 07 0. 20312-12 100100 251110 1140275 1: 305-11 023330 300.00 5t'.t'!5 1.41107. 1.30230 0.22045 7 g mi 5; 14 5757 913 37 It can be S tfifl fl'OflI Table I that f0! all VQIUCS Of M0, 30000.00 1414.000 -11. 3000 7 .3 11-15% 1 1 is substantially 1.4, while in the longitudinal vibrators of my above-mentioned copending application, 1p is substantially 1.6. Thus the torsional transformers are slightly less efiicient producers of large particle velocity for given maximum strain than are the longitudinal transformers. They are, however, still much better than a stepped torsional transformer, consisting of two quarterwave cylinders of substantially differing diameters having a common longitudinal axis and joined by a fillet, would be. Such a stepped transformer would have, as discussed in my copending application for the longitudinal case, a value of :p0.8.

Although the value of (p is only slightly smaller for the torsional than for the comparable longitudinal transformer, care must be taken in comparing the maximum particle velocities of which the two types of vibrator are capable. This velocity is equal to pS c, Equation 15. The speed of sound, 0, is smaller for shear waves than for extensional waves in the ratio 1:[2(1+u)] where v is Poissons' ratio. If the material is liable to failure by a plastic process such as yield or fatigue, as discussed by applicant in an article published in the periodical Nature," volume 188, pages 1183 and 1184, London, December 31, 1960, then the maximum shear stress is a good criterion of the limit at which one may expect failure. The maximum (engineering) shear strain is then related to the tensile strain in plane stressas (l+u):l. Therefore, for such materials:

m/ p) torsional rn/ P) t-xtens innul If, on the other hand, the material is essentially brittle, so that the maximum principal tensile strain is the criterion of the limit at which one may expect failure, then the maximum permissible (engineering) shear strain in pure shear is twice the maximum permissible tensile strain in plane stress and therefore m/ P)toraional for m/1 extenaiunui Engineering shear strain is discussed, for example, in the above-mentioned book by W. P. Mason, entitled Physical Acoustics and'the Properties of Solids, at page 356. Similarly, elastic fracture and flow is discussed, for example, in a book of J. C. Jaeger directed to this subject, published by John Wiley & Sons, New York, second edition, 1962.

Thus, a Fourier torsional amplitude transformer with p=l.4, if made of a ductile metal with v=0.3, is capable of 30 percent smaller maximum particle velocity than is the corresponding longitudinal transformer with =l.6. But if it is made of a brittle material with 11:03, the torsional transformer is capable of 10 percent greater particle velocity than the longitudinal. In assessing this it should be remembered that, because a is substantially 0.6%, the torsional transformer is correspondingly smaller than the comparable longitudinal transformer. This is illustrated in FIG. 2 of the accompanying drawing where members 14 and 18 represent the size and contour of comparable torsional and longitudinal transformers, respectively.

A further important object of the invention is, as mentioned above, to produce transformers of high magnification that are not too slender. In FIGS. 3 and 4 the performance in this respect is shown by characteristic 20 of FIG. 3 for angular magnification, M and for magnification of surface-particle velocity, M by characteristic former for angular magnification. FIG. 4, characteristic 32, illustrates the relative performance of either a torsionally resonant or longitudinally resonant exponentially tapered transformer for surface-particle velocity or linear magnification. It can be shown that an exponential transformer has M =R and M =R lt is, however, clear, from the characteristics as above noted, that, except at very small magnifications, the torsional Fourier transformers are very much less slender, and consequently much stiffer, than the comparison (exponential) type. This is emphasized in FIGS. 1 and 2, as noted hercinabovc, where profiles or longitudinal contours of five transformers of magnification 700 are drawn. It is clear from the stubbiness of the Fourier transformer with M,,=700, characteristic 10 of FIG. 1, that transformers of this type having extremely large angular magnifications are perfectly practical to make and operate.

For surface magnification (FIGS. 4 and 2), where the exponential transformer becomes quite unmanageable at high magnification, since the larger portion of its length becomes impracticably slender, the torsional Fourier transformer has to compete with the Fourier longitudinal transformer. The latter has the advantage, from the point of view of slenderness, that the wavelength for longitudinal waves is greater than that for torsional waves, which allows larger diameters to be used for a given frequency.

Another criterion of slenderness is the normalized flexural compliance, C which is defined as the inverse of stiffness in my above-mentioned copending application. Allowance must, of course, be made for the difference in wavelength between torsion and extension. In FIG. 5, the flexural compliance, C is plotted against magnification, assuming Poissons ratio to be 0.3. Characteristics 40, 42 and 44 of FIG. 5 are, as designated, those for the Fourier torsional angular magnification, Fourier torsional linear magnification, and Fourier longitudinal types of amplitude transformers, respectively. Thisfigure shows not only how very stiff the transformers of angular velocity are, but also how much stiffer the torsional trans formers of particle velocity are than their longitudinal counterparts, except at very small magnification. Thus, for example, the torsional transformer with M =700 is about 20 times stiffer than the longitudinal one with M=700.

Numerous and varied modifications and rearrangements of the above described specific embodiments may readily be devised by those skilled in the art Without departing from the spirit and scope of the principles of the invention. The embodiments described are illustrative only and are not to be taken as limiting the invention.

What is claimed is:

1. A mechanical transformer for magnifying the amplitude of a repetitive, vibratory, torsional physical displacement by a factor M,,, M,, having a value exceeding 2, said transformer comprising an elongated acoustically resonant member of constant cross-sectional shape or contour having one end larger than the other end, a length I less than a wavelength A for torsional waves of the members resonant frequency propagated along a uniform member of the same cross-sectional shape and the same material, a maximum transverse dimension not greater than one-half of said wavelength, the torsional rigidity K of the member varying between the larger and smaller ends of the member in accordance with the integral of the relation 30 of FIG. 4, respectively. For comparison, character istic 34 of FIG. 4 is shown for the longitudinal Fourier transformer for linear magnification and characteristic 22 of FIG. 3 for the torsionally resonant exponential transwhere 1 i 1 2 where x represents the distance, along the member from 4. The transformer of Claim 1 in which {3 has a value its larger end, of the cross section for which the torsional belwen -0-23 and -0-35. rigidity K is being instantly determined and fig is a con- Thc transformc! 0f c1111111 1 111 which 54 1S Z510- Zigl'ltQiiChOlCfi having a value between minus one and 5 References Cited y the Examiner 2. The transformer of claim 1 in which m is chosen to TED TAT S PATENTS give substantially the lowest ration of maximum to mini- 3 31,515 5 9 Mason 7 7 X mum cross-sectional areas for the selected value of 3. The transformer of claim 2 in which [1 has a value BROUGHTON DURHAM Exammer' hetween -0.28 and --0.35. 10 D. H. THIEL, Assistant Examiner. 

1. A MECHANICAL TRANSFORMER FOR MAGNIFYING THE AMPLITUDE OF REPETITIVE, VIBRATORY TORSIONAL PHYSICAL DISPLACEMENT BY A REFRACTOR M0, M0 HAVING A VALUE EXCEEDING 2, SAID TRANSFORMER COMPRISING AN ELONGATED ACOUSTICALLY RESONANT MEMBER OF CONSTANT CROSS-SECTIONAL SHAPE OR CONTOUR HAVING ONE END LARGER THAN THE OTHER END, A LENGTH LT LESS THAN A WAVELENGTH $T FOR TORSIONAL WAVES OF THE MEMBER''S RESONANT FREQUENCY PROPAGATED ALONG A UNIFORM MEMBER OF THE SAME CROSS-SECTIONAL SHAPE AND THE SAME MATERIAL, A MAXIMUM TRANSVERSE DIMENSION NOT GREATER THAN ONE-HALF OF SAID WAVELENGTH, THE TORSIONAL RIGIDITY K1 OF THE MEMBER VARYING BETWEEN THE LARGER AND SMALLER ENDS OF THE MEMBER IN ACCORDANCE WITH THE INTEGRAL OF THE RELATION 